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June 24, 2013June 24, 2013

Reality = Math?

Introduction

A long time ago I read an introduction to a book in which the author suggested (with some seriousness) that reality was made of numbers, or at least math. He then offered an example to show that people considered math more fundamental than anything. The following is a paraphrase of that example:

Imagine you have one apple on a table. You then take an apple out of a bag and put it on the table next to the first apple. How many apples are now on the table? There should be two. But imagine there are now three. What do you do?

Most people would check for trickery or assume they miscounted the number of apples originally on the table. If this keeps happening, they may assume some weirdness like teleporting apples. They may even start to doubt their eyes or sanity. What they would not do is doubt that 1+1=2. It would not even occur to them to doubt that.

Math is so fundamental that it trumps physics and even our very perceptions. Let’s look at another example of how fundamental this attitude towards the primacy of math is.

The Engineer and the Machine

Let’s say a factory engineer observes the reliability of a machine after so many hours of operation. She starts by collecting data on the defect rate at different points in time. For example, she may find the following:

Time. Defect
3. 0.006
5. 0.01
8. 0.016

Armed with this data, and assuming Time determines Defects, she now wonders if there is a rule she can apply to predict unobserved defect rates. For example, she never measured the defect rate at 15,000 hours, or even at 1.7 hours and would like to predict those.

In assuming an answer to this question, she is assuming there is a rule that relates Time to Defects. Naturally, this rule is assumed to be mathematical. Not only is the machine assumed to follow a mathematical rule, but the defect prediction is assumed to be a mathematical problem that is independent of the machine. In short, the problem becomes to find the mathematical structure of the data.

To find the defect rate, she simply finds the relation that governs the data. There are tons of mathematical techniques that do this, but this example was meant to be simple so the pattern may be obvious. In this case, she finds Defect = 0.002 x Time

The Subject is Irrelevant. The Math is Not

How was this found? It had nothing to do with the machine. It was purely the relations obtaining among the data. We could have been talking about cola consumption and violent crime, and the process and formula would have been the same. The engineer could have delivered the data to a mathematician, removed all explanations, and simply asked for a formula. Heck, she could have plugged those six numbers into any curve fit program, and gotten that formula.

What’s more, anyone armed with the formula could predict the defect rate, and would need to know nothing about machines to do it. In fact, if we called the formula f, and simply asked for the value of f at 15, this person could provide an answer to the machine defect rate and at no point would mention of a machine have entered into it. The person would have provided an answer to the machine’s defect rate without knowing it. This kind of knowledge does not sit well with some people. For an anlgous situation, check out Searle’s Chinese Room thought experiment.

In short, once the data has been gathered, this turned into a purely mathematical problem.

So when we seek certain types of knowledge — possibly most types of predictive knowledge — we are seeking nothing more than the mathematical structure of data, with no reference to the subject or source of this data. Put another way, most non-trivial knowledge is mathematical knowledge, and not knowledge of a subject per se.

Is reality math?

Bye Bye Data

It gets better or worse, depending in your perspective, for the mathematical structure not only trumps the subject matter, but it may even trump the data itself! Let’s start by visualizing a formula by plotting a point for each data pair. Once the points are plotted, we can try to fit a shape to all the points and this shape then becomes a picture of our function. Here is a picture of a different process:

Courtesy of Wikipedia

The curve passes through all the points and we are good.

But what if the data were different? For example, take this plot:

Courtesy of Wikipedia

Notice how the line does not connect with the dots. Connecting the dots would produce a crooked, disjointed line, and a more complex formula. So what do we do? Well standard practice is to go with the straight line above, and the resulting function, even though it does NOT fit all the data. We assume the data is “noisy”. That is, we assume the observations do correspond to the line, but due to imprecise measurements, observation error, maybe even the operation of a hidden variable, it wasn’t measured as such. In some extreme cases, we may even discard some points (or re-assign them) as outliers.

Let that sink in. When the data does not match the a priori mathematical assumption, we basically reject the data. Again, we doubt our perceptions before we doubt the math. The assumption is that reality behaves according to smooth and relatively simple mathematical laws, and we refuse to let reality get in the way of this assumption.

Closing Thoughts

Now the situation is not as cut and dried. Sometimes with enough anomalous data, we have to take notice. Interestingly enough, when we do, its usually to find another smooth, simple mathematical function that fits this data. Also in some cases, knowledge of the subject helps, as certain classes of processes are known to have specific classes of formulas, and this can guide us in fitting a formula and even in deciding if any data is to be accepted or rejected. In some cases, we may even grit our teeth and accept disjointed functions (splines). But the evidence has to be compelling. That is, we assume smooth, simple mathematical reality unless there’s compelling evidence otherwise. And if we grudgingly accept a more complex model, we assume there is a simpler one we haven’t discovered yet — that basically this model isn’t “really” what is happening.

But is there a chicken and egg problem here? Is it that math is supreme, or is it that math is merely an abstraction of the parts of reality that matter to us? If it is the latter, then arguing reality is mathematical is tantamount to arguing that reality behaves in ways previously observed. Even in cases where abstract math has shown surprising real life applications, we can argue that even the most abstract math is ultimately derived from a study of the concrete, being as it is a study of relations or patterns. Since reality is all about relations or patterns, it would stand to reason they should obey these laws.

Questions

Is there anything special about math?
How do you explain the effectiveness of math in describing reality?
Can you point to a priori math concepts you encountered?
Can you think of areas where math failed?

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19 thoughts on “Reality = Math?”

Great post on two of my favourite subjects maths and the (origin) of reality. As a language, maths is by far the most rigorous in its description of our reality it is tempting to think that it is somehow innate and the source of it all. However, mathematics is developed by human beings and even if it does a marvelous job in allowing us to reason accurately it has a human limitation built in: duality. All maths revolves around equations. An equation is an object with two sides. Before and after states or two forms of the same thing. That separation creates the very notion of things vs other things and that is a human perspective at the core of all language. I believe that the duality of mathematics is its limitation in describing all reality and that would exclude the ultimate a priori.

For a while I have been playing with the notion of “non-equitive” mathematics, i.e. objects without equal signs. I couldn’t make useful sense of that but I haven’t given up hope that it may be made to work and provide a step forward in describing it all.

Remove the equation sign from the toolkit. What you are left with is objects like x^2 or x+y. What is the use of an equation sign? It gives a context to such objects. x^2=2 positions the parabola. Without the equation it is still a parabola but it is everywhere and nowhere. Stripping such objects from their context and bringing them back their basic constituents leaves us with the most basic mathematical identities (particles). 1,0, pi,e, + etc. (Euler’s identity without the equal sign).

What is the point of such a reduction? Perhaps it is to show us that the only way to turn fundamental mathematical particles in something useful, we need operations AND equal signs. That would make the dualistic equal sign the most creative force in mathematics. Who would have thought…

That begs the question: what the hell is an equal sign if it is so powerful?

I have thought hard about dualities in general and concluded that not only are they a human perspective, they are also a false view of a situation. The opposite sides of a duality are two extreme values on the same spectrum. Ergo, the equal sign is an shortform for a mathematical spectrum on which two side of an equation exist. That spectrum is a continuum and there is no dualistic separation.

Taking that thought further would make every equal sign in every equation a different spectrum.

Have I confused you enough?

One last thought. What is the basic tenet of eastern religion? Things are not this or that. Things are. A thing IS. A thing =

You’d both a genius and a medium if you could make sense of that 🙂 (which doesn’t mean that if you are a genius and a medium that it makes sense but it helps) I owe a few people a post on my thoughts on the spectrum of dualism. It would certainly help here!

Well the equation part I understand; it’s another way of looking at fixing a function at a value. But that’s another good way of thinking about it.

Going back to your previous comment, it’s treating the equation as an algorithm and plugging in data for that algorithm. The algorithm denotes a whole class of processes or potential values, but actually getting a subset of those values requires either plugging in data or setting it equal to something. Both restrict it from different sides so to speak.

Hi there BR!
It’s good to see you blogging again. 🙂 I’m happy to say that I can follow your reasoning and I find your first example, about the apples, very interesting. I couldn’t imagine anyone looking to math itself as an explanation for the problem and once you mentioned it I couldn’t imagine why.
I think many people would be surprised to read this, or be even more attached to math and reject it altogether, saying it was a wrong example, somehow.
I’m currently spending some time thinking about the concept of flux and I’m having similar experiences. That is, on the face of it you can say: “Everything around us and we humans are in a state of flux.” and that sounds reasonable. Until you start to think about what that really means.
So thank you for writing this and making me think about it!

Thanks for reading. I think a lot of this ties in with the notion of flux if you are willing to take flux as an expression of change. That is, if you stop thinking in terms of things and start thinking in terms of relation and process, then math and even language take on a new applicability.

An enjoyable post, thank you. And it struck some chords with me. It’s not original to point out that what we think of as the embodiment of solidity, all those physical objects around us, are anything but. The solidity is a useful illusion we invent for ourselves. Solidity is, in any case, an odd concept: if something were literally solid, it would have to be infinitely dense. So an atomic theory seems to be logically inevitable, as well as an empirical finding: I suspect that thoughts such as these were in the minds of the ancient Greek atomists such as Democritus.
So what does this have to do with maths (as we call it in the UK) – ? Our direct experience of objects may not tell us what they are ‘really like’, but in quantum physics we have a way of describing how they are, and it’s exclusively mathematical. There’s no point in speculating about what atoms or electrons etc. are ‘really like’, imagining that we were able to shrink ourselves down and examine them. Our bodies and perceptual apparatus operate necessarily at a macro level – no such experience could ever be possible. We can only describe the ultimate constituents of matter in terms of equations (laws), and numbers (positions and momenta). And we know all about the famous uncertainty that surrounds those.

I think the algorithmic approach to maths clears a lot of issues like this. If we consider maths to be anything that can be programmed, then reality has to be mathematical almost by definition. The only other possibility would be for it to be non-computable like Chaitin’s Omega number, which in a way would still be maths (it is defined mathematically). Not only that, but in an irreducible world where no algorithmic compression is possible, parts of the world would still be amenable to compression. For example, our reality could be a part of an irreducibly complex world which is itself not irreducibly complex and thus subject to being expressed as an algorithm. This view of mathematics leaves very little room for anything non-mathematical. In fact, I find myself incapable of finding an example!